On Carlier's inequality
Optimization and Control
2025-07-31 v1
Abstract
The Fenchel-Young inequality is fundamental in Convex Analysis and Optimization. It states that the difference between certain function values of two vectors and their inner product is nonnegative. Recently, Carlier introduced a very nice sharpening of this inequality, providing a lower bound that depends on a positive parameter. In this note, we expand on Carlier's inequality in three ways. First, a duality statement is provided. Secondly, we discuss asymptotic behaviour as the underlying parameter approaches zero or infinity. Thirdly, relying on cyclic monotonicity and associated Fitzpatrick functions, we present a lower bound that features an infinite series of squares of norms. Several examples illustrate our results.
Cite
@article{arxiv.2206.14872,
title = {On Carlier's inequality},
author = {Heinz H. Bauschke and Shambhavi Singh and Xianfu Wang},
journal= {arXiv preprint arXiv:2206.14872},
year = {2025}
}